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Partition of a graph whose components are reachable from all vertices
graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of a directed graph form a
Strongly_connected_component
Topics referred to by the same term
Connected component may refer to: Connected component (graph theory), a set of vertices in a graph that are linked to each other by paths Connected component
Connected_component
Maximal subgraph whose vertices can reach each other
graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition
Component_(graph_theory)
Algorithmic application of graph theory
Connected-component labeling (CCL), connected-component analysis (CCA), blob extraction, region labeling, blob discovery, or region extraction is an algorithmic
Connected-component_labeling
Topological space that is connected
collection of connected subsets such that each contains x {\displaystyle x} will once again be a connected subset. The connected component of a point x
Connected_space
Basic concept of graph theory
edge. A graph is connected if and only if it has exactly one connected component. The strong components are the maximal strongly connected subgraphs of a
Connectivity_(graph_theory)
Graph algorithm
Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph
Tarjan's strongly connected components algorithm
Tarjan's_strongly_connected_components_algorithm
Large connected component of a random graph
In network theory, a giant component is a connected component of a given random graph that contains a significant fraction of the entire graph's vertices
Giant_component
Property of topological spaces
connected and locally path connected. A space is locally connected if and only if for every open set U, the connected components of U (in the subspace topology)
Locally_connected_space
Space which has no holes through it
connected) open set has a connected extended complement exactly when each of its connected components is simply connected. Informally, an object in our
Simply_connected_space
Mathematical concept
pieces, each piece is usually called a component (or connected component). A topological space is said to be connected if it is not the union of two disjoint
Connectedness
Maximal biconnected subgraph
theory, a biconnected component or block (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into
Biconnected_component
Tree data structure that partitions a 2D area
one unique label for each connected component label the black pixels with the label associated with their connected component To simplify the discussion
Quadtree
Topics referred to by the same term
color components Component (group theory), a quasi-simple subnormal sub-group Connected component (graph theory), a maximal connected subgraph Connected component
Component
Concept in group theory
the identity component of a group G (also known as its unity component) refers to several closely related notions of the largest connected subgroup of
Identity_component
Trail in a graph that visits each edge once
Euler tour in each connected component of G and then orienting the edges according to the tour. Every Eulerian orientation of a connected graph is a strong
Eulerian_path
Method of finding a directed graph's strongly connected components
Kosaraju's algorithm) is a linear time algorithm to find the strongly connected components of a directed graph. Aho, Hopcroft and Ullman credit it to S. Rao
Kosaraju's_algorithm
Partition of vertices of a directed graph
weak components were defined in a 1972 paper by Ronald Graham, Donald Knuth, and (posthumously) Theodore Motzkin, by analogy to the strongly connected components
Weak_component
Graphical representation of a computer program or algorithm
strongly-connected components. The strongly-connected components of a CFG form a directed acyclic graph called the SCC tree, where component A has an
Control-flow_graph
Pathological embedding of the sphere in 3D space
{\displaystyle \mathbb {R} ^{3}} is not simply connected. Specifically, the fundamental group of the unbounded component of R 3 ∖ f ( S 2 ) {\displaystyle \mathbb
Alexander_horned_sphere
Measure of the structural complexity of a software program
number of connected components. An alternative formulation of this, as originally proposed, is to use a graph in which each exit point is connected back to
Cyclomatic_complexity
Least-weight tree connecting graph vertices
(not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are many
Minimum_spanning_tree
Theorem in topology
{\displaystyle C} is the boundary of each component. In contrast, the complement of a Jordan arc in the plane is connected. The Jordan curve theorem was independently
Jordan_curve_theorem
for 2-vertex-connected, but sometimes includes K2 though it is not 2-connected. See connected; for biconnected components, see component. binding number
Glossary_of_graph_theory
Algebraic structure associated with a topological space
one-dimensional sphere S 1 {\displaystyle S^{1}} is a circle. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes
Homology_(mathematics)
Weighted tree representing s-t cuts of a graph
is a connected component in T ∖ X } . {\displaystyle S=\{S_{C}\mid C{\text{ is a connected component in }}T\setminus X\}.} Contract the components to form
Gomory–Hu_tree
Inverse of a finite difference
C} , on that connected component. For functions with isolated poles, distinct components must give different branches: the native component provides a pole-free
Indefinite_sum
Data structure that maintains info about the connected components of a graph
data structure that dynamically maintains information about the connected components of a graph. The set V of vertices of the graph is fixed, but the
Dynamic_connectivity
Academic field
all others. Giant Component: A single connected component which contains most of the nodes in the network. Weakly Connected Component: A collection of
Network_science
Type of group in mathematics
group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup
Orthogonal_group
Branch of topology
The space X is said to be path-connected (or pathwise connected or 0-connected) if there is at most one path-component; that is, if there is a path joining
General_topology
Graph with at most one cycle per component
graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges
Pseudoforest
Abstract strategy board game
connected component of one of the red edges, i.e. all the red hexagons directly or indirectly connected to that red edge. The concept of a connected component
Hex_(board_game)
Type of computer science algorithm
to know if two vertices in a graph of n vertices are in the same connected component of the graph, there is no known simple, deterministic, in-place algorithm
In-place_algorithm
Harold N.; Somenzi, Fabio (January 2006), "An algorithm for strongly connected component analysis in n log n symbolic steps", Formal Methods in System Design
Quotient_graph
Minimum spanning forest algorithm that greedily adds edges
spanning forest is composed of a minimum spanning tree for each connected component. This algorithm was first published by Joseph Kruskal in 1956, and
Kruskal's_algorithm
Type of relation for subsets of a topological space
separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological
Separated_sets
Logic problem, AND of pairwise ORs
either by a method based on backtracking or by using the strongly connected components of the implication graph. Resolution, a method for combining pairs
2-satisfiability
Method for finding minimum spanning trees
finds minimum spanning trees in connected graphs. However, running Prim's algorithm separately for each connected component of the graph, it can also be
Prim's_algorithm
Computer science algorithm
or BFS), then the algorithm must be called at least once for each connected component of the graph. This is easily accomplished by iterating through all
Graph_traversal
Directed graph with no directed cycles
DAG, called its condensation, by contracting each of its strongly connected components into a single supervertex. When the graph is already acyclic, its
Directed_acyclic_graph
Type of graph
graph. Every 2-connected graph can be constructed inductively by adding paths to a cycle (Diestel 2016, p. 59). Biconnected component Eric W. Weisstein
Biconnected_graph
Types of electrical circuits
Two-terminal components and electrical networks can be connected in series or parallel. The resulting electrical network will have two terminals, and
Series_and_parallel_circuits
Method for finding minimum spanning trees
repetition of this process reduces the number of trees, within each connected component of the graph, to at most half of this former value, so after logarithmically
Borůvka's_algorithm
Overview of and topical guide to algorithms
algorithm Borůvka's algorithm Connected component (graph theory) Strongly connected component Tarjan's strongly connected components algorithm Maximum flow problem
Outline_of_algorithms
complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property
Planar_Riemann_surface
Edge whose deletion would disconnect a graph
graph's number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge
Bridge_(graph_theory)
Matrix representation of a graph
0-eigenvector if and only if it has a bipartite connected component (isolated vertices being bipartite connected components). This can be shown as x T Q x = x T
Laplacian_matrix
Category where every morphism is invertible; generalization of a group
called its connected components (possibly with different groups G {\displaystyle G} and sets X {\displaystyle X} for each connected component). In category-theoretic
Groupoid
Type of function in mathematics
its domain, then f {\displaystyle f} is zero everywhere on the connected component containing the accumulation point. In other words, if ( r n ) {\displaystyle
Analytic_function
Discrete device in an electronic system
An electronic component is any basic discrete electronic device or physical entity part of an electronic system used to affect electrons or their associated
Electronic_component
Every path-connected space is connected. Path-connected component A path-connected component of a space is a maximal nonempty path-connected subspace.
Glossary_of_general_topology
Method of data analysis
Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data
Principal_component_analysis
Topics referred to by the same term
principal components analysis Component analysis (statistics), any analysis of two or more independent variables Connected-component analysis, in graph theory
Component_analysis
Connectivity measure in graph theory
at v. If G is not strongly connected, then r(G) is equal to the maximum cycle rank among all strongly connected components of G. The tree-depth of an
Cycle_rank
Recursively splitting a graph into subsets of nodes
called modules. A module is a generalization of a connected component of a graph. Unlike connected components, however, one module can be a proper subset of
Modular_decomposition
Undirected, connected, and acyclic graph
graph or equivalently a disjoint union of trees. Trivially so, each connected component of a forest is a tree. As special cases, the order-zero graph (a
Tree_(graph_theory)
coproduct) of a collection of connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.
Connected_category
Theorem in complex analysis
If A is a set containing at least one complex number from every connected component of C ∪ { ∞ } ∖ K {\displaystyle \mathbb {C} \cup \{\infty \}\setminus
Runge's_theorem
French mathematician
critical point, the probability of having two vertices in the same connected component of the lattice would decay exponentially with separation distance
Hugo_Duminil-Copin
Roughly, the number of k-dimensional holes on a topological surface
number of connected components. Therefore, the "zero-th" Betti number b0(G) equals |C|, which is simply the number of connected components. The first
Betti_number
Copy of a directed graph with redundant edges removed
strongly connected components of G that are connected by an edge in the condensation, x is any vertex in component X, and y is any vertex in component Y. The
Transitive_reduction
Proposed legislation in the United States
pdf "Moreno, Slotkin Bill to Ban Chinese Vehicles, Connected Components From U.S. Market". Senator Bernie Moreno. Retrieved April 30, 2026
Connected Vehicle Security Act
Connected_Vehicle_Security_Act
Two closely related models for generating random graphs
c ′ {\displaystyle p'_{c}} a giant connected component of order n exists. The relative size of the giant component, P∞, is given by P ∞ = p ′ [ 1 − exp
Erdős–Rényi_model
Cellular automaton neighborhood consisting of eight adjacent cells
tracing algorithm is Input: A square tessellation, T, containing a connected component P of black cells. Output: A sequence B (b1, b2, ..., bk) of boundary
Moore_neighborhood
Group that is also a differentiable manifold with group operations that are smooth
Gcon for the connected component of the identity Gsol for the largest connected normal solvable subgroup Gnil for the largest connected normal nilpotent
Lie_group
Point in a topological space
subspace is disconnected, or, equivalently, every connected component is a single point). If X is connected and X ∖ { p } {\displaystyle X\setminus \{p\}}
Dispersion_point
Trail in which only the first and last vertices are equal
into strongly connected components, cycles only exist within the components and not between them, since cycles are strongly connected. For directed graphs
Cycle_(graph_theory)
Graph algorithm
In graph theory, the strongly connected components of a directed graph may be found using an algorithm that uses depth-first search in combination with
Path-based strong component algorithm
Path-based_strong_component_algorithm
Type of random graph
the set of open bonds, then an open cluster or FK cluster is any connected component in A ( ω ) {\displaystyle A(\omega )} union the set of vertices.
Random_cluster_model
Set associated with a complex-valued polynomial
Hull}}\{j\in A\mid a_{j}\neq 0\}.} Then Any amoeba is a closed set. Any connected component of the complement R n ∖ A p {\displaystyle \mathbb {R} ^{n}\setminus
Amoeba_(mathematics)
Whether one vertex can be reached from another in a graph
the connected components of the graph. Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component;
Reachability
(pseudo-)Riemannian manifold whose geodesics are reversible
symmetric space is the quotient G / H of a connected Lie group G by a closed subgroup H that is (a connected component of) the invariant group of an involution
Symmetric_space
Combinatorial optimization method for pseudo-Boolean functions
negation) are separated by the minimum cut of the graph in two different connected components, then the optimal value for such variable is well defined, otherwise
Quadratic pseudo-Boolean optimization
Quadratic_pseudo-Boolean_optimization
Term in mathematics
genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety
Jacobian_variety
Tree which includes all vertices of a graph
or equivalently a subgraph consisting of a spanning tree in each connected component of the graph. To avoid confusion between these two definitions, Gross
Spanning_tree
Form of a matrix
image of the exponential map of a Lie algebra always lies in the connected component of the Lie group that contains the identity element. In the case
Skew-symmetric_matrix
Family of random graph models
giant components is vanishing. This means that in the sparse regime, the model consist of one giant component (if any) and multiple connected components of
Configuration_model
mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents
End_(topology)
Set of graph nodes which separate a given pair of nodes if removed
the removal of S from the graph separates a and b into distinct connected components. Consider a grid graph with r rows and c columns; the total number
Vertex_separator
Abstract data type in computer science
all of the nodes in a given connected component. Both start with an arbitrary node, the "root". Strongly connected components can also be found using graph
Graph_(abstract_data_type)
Characterization of graphs with perfect matchings
Since each vertex in U can be in this relation with at most one connected component (because of it being matched at most once in a perfect matching)
Tutte's theorem on perfect matchings
Tutte's_theorem_on_perfect_matchings
Abstraction of unicyclic subgraphs
sets of pseudoforests of G, that is, the edge sets in which each connected component contains at most one cycle. The bicircular matroid was introduced
Bicircular_matroid
Generalization of a scheme
equivalence relation over each connected component of U (i.e. for all x, y belonging to the same connected component of U, we have xRy if and only if
Algebraic_space
{\displaystyle C(M,N)} , any path-connected component of W s , p ( M , N ) {\displaystyle W^{s,p}(M,N)} and any path-connected component of C ( M , N ) {\displaystyle
Sobolev_mapping
Lie group of Lorentz transformations
Lie group that is not connected. The four connected components are not simply connected. The identity component (i.e., the component containing the identity
Lorentz_group
Graph which remains connected when k or fewer nodes removed
into a disjoint union of 1-connected components. 1-connected graphs decompose into a tree of biconnected components. 2-connected graphs decompose into a
Vertex_connectivity
Algorithm to search the nodes of a graph
Finding connected components. Topological sorting. Finding 2-(edge or vertex)-connected components. Finding 3-(edge or vertex)-connected components. Finding
Depth-first_search
Edges that hit all cycles in a graph
strongly connected component of the given graph, and to break these strongly connected components down even farther to their biconnected components by splitting
Feedback_arc_set
though the algorithm is designed for connected graphs, it can be applied individually to each connected component of a graph with the same running time
Seidel's_algorithm
Figure-eight-shaped curve
{\displaystyle y^{2}(y^{2}-a^{2})=x^{2}(x^{2}-b^{2})} in which one connected component has a figure-eight shape, Watt's curve, a figure-eight shaped curve
Lemniscate
Concept in complex analysis
antiderivative of the zero function if and only if it is constant on each connected component of U {\displaystyle U} (those constants need not be equal). This
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Independent set which is not a subset of any other independent set
each connected component always enters I, so there is always some progress. In particular, in the worst-case of the previous algorithm (n/2 connected components
Maximal_independent_set
Double cover Lie group of the special orthogonal group
not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact SO(p)
Spin_group
strongly connected component of the given graph. Letting s {\displaystyle s} denote the number of source vertices in the condensation (strongly connected components
Strong connectivity augmentation
Strong_connectivity_augmentation
Analog of the continuous Laplace operator
ones and zeros, where each connected component corresponds to an eigenvector with ones at the elements in the connected component and zeros elsewhere. The
Discrete_Laplace_operator
Concept in graph theory
such that every connected component that remains after removing the vertices of P {\displaystyle P} from G {\displaystyle G} is connected back to P {\displaystyle
Tutte_path
Concept in complex dynamics
Every connected component of the closure of the escaping set is unbounded. The escaping set always has at least one unbounded connected component. The
Escaping_set
Subset which is both open and closed
a union of (possibly infinitely many) connected components of X {\displaystyle X} . If all connected components of X {\displaystyle X} are open (for instance
Clopen_set
graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an
Moral_graph
CONNECTED COMPONENT
CONNECTED COMPONENT
Boy/Male
Hindu
Collected
Boy/Male
Hindu
Connected, United
Boy/Male
Tamil
Sanyukt | ஸஂயà¯à®•à¯à®¤
Connected, United
Sanyukt | ஸஂயà¯à®•à¯à®¤
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Connected
Girl/Female
Sikh
Associated, Connected
Girl/Female
Tamil
Collected
Boy/Male
Arabic, Muslim
Joined; Arrived; Connected
Girl/Female
Tamil
Yuktatma | யà¯à®•à¯à®¤à®¾à®¤à®®à®¾à®‚
Self connected
Yuktatma | யà¯à®•à¯à®¤à®¾à®¤à®®à®¾à®‚
Girl/Female
Australian, Celtic, Irish
Connected to Irish Mythology
Girl/Female
Celtic
Contented.
Boy/Male
Tamil
Collected
Boy/Male
Gujarati, Indian
Connected
Boy/Male
Tamil
Collected
Boy/Male
Hindu
Collected
Girl/Female
Muslim
Collected
Boy/Male
Native American
Conceited.
Boy/Male
Tamil
Attached, Connected
Girl/Female
Hindu
Self connected
Boy/Male
Hindu
Attached, Connected
Girl/Female
Tamil
Collected
CONNECTED COMPONENT
CONNECTED COMPONENT
Boy/Male
Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Sikh, Telugu
Diamond; Wealth
Boy/Male
Australian, Biblical
Wicked; Worthless
Girl/Female
Celtic
Songbud.
Boy/Male
Hindu, Indian
King of Heart
Boy/Male
Tamil
Name of a prophet, A nabee
Female
English
Pet form of English Hannah, HANNY means "favor; grace."
Boy/Male
American, British, English, Teutonic
Friend from the North
Biblical
earthy; red; bloody things
Boy/Male
Tamil
Bramhanand | பà¯à®°à®®à¯à®¹à®¾à®¨à®‚த
Happiness for knowledge
Girl/Female
Hindu
The youngest, Girl, Maiden, Daughter, The virgin Goddess
CONNECTED COMPONENT
CONNECTED COMPONENT
CONNECTED COMPONENT
CONNECTED COMPONENT
CONNECTED COMPONENT
a.
Having an overweening opinion of one's own powers, attainments; vain; conceited.
a.
Convicted by one's own consciousness, knowledge, avowal, or acts.
imp. & p. p.
of Contest
a.
Connected with, or serving to connect, three channels or pipes; as, a three-way cock or valve.
imp. & p. p.
of Confect
v. i.
To be connected.
a.
Separate; unconnected, or imperfectly connected; as, detached parcels.
imp. & p. p.
of Convert
a.
United; connected; associated.
adv.
Closely connected or related.
a.
Mutually contrived or planned; agreed on; as, concerted schemes, signals.
v. i.
To join, unite, or cohere; to have a close relation; as, one line of railroad connects with another; one argument connect with another.
imp. & p. p.
of Connect
imp. & p. p.
of Correct
a.
Content; easy in mind; satisfied; quiet; willing.
p. a.
Unconnected; not united or associated; distinct; -- said of things that have not been connected.
adv.
In a connected manner.
a.
Not connected; disconnected.
imp. & p. p.
of Convict
n.
One who, or that which, connects